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Homedependence relation
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dependence relation
Let $X$ be a set. A (binary) relation $\prec$ between an element $a$ of $X$ and a subset $S$ of $X$ is called a dependence relation, written $a\prec S$, when the following conditions are satisfied:
1. if $a\in S$, then $a\prec S$;
2. if $a\prec S$, then there is a finite subset $S_{0}$ of $S$, such that $a\prec S_{0}$;
3. if $T$ is a subset of $X$ such that $b\in S$ implies $b\prec T$, then $a\prec S$ implies $a\prec T$;
4. if $a\prec S$ but $a\not\prec S\{b\}$ for some $b\in S$, then $b\prec(S\{b\})\cup\{a\}$.
Given a dependence relation $\prec$ on $X$, a subset $S$ of $X$ is said to be independent if $a\not\prec S\{a\}$ for all $a\in S$. If $S\subseteq T$, then $S$ is said to span $T$ if $t\prec S$ for every $t\in T$. $S$ is said to be a basis of $X$ if $S$ is independent and $S$ spans $X$.
Remark. If $X$ is a nonempty set with a dependence relation $\prec$, then $X$ always has a basis with respect to $\prec$. Furthermore, any two bases of $X$ have the same cardinality.
Examples:

Let $V$ be a vector space over a field $F$. The relation $\prec$, defined by $\upsilon\prec S$ if $\upsilon$ is in the subspace spanned by $S$, is a dependence relatoin. This is equivalent to the definition of linear dependence.

Let $K$ be a field extension of $F$. Define $\prec$ by $\alpha\prec S$ if $\alpha$ is algebraic over $F(S)$. Then $\prec$ is a dependence relation. This is equivalent to the definition of algebraic dependence.
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